Alan Alda is absolutely the sexiest 70+ year old out there. He’s also a fabulous communicator of science — perhaps, along with Neil deGrasse Tyson, the best of the present generation.
There’s a video that has gone viral and concerns itself with the equation
A couple of students from my Calculus 2 class last semester sent me the following e-mail, which makes me smile for several reasons:
We are rather skeptical, but do not really understand explanations of the “proofs” others have provided. Since we consider you the expert on convergence and divergence we were hoping you knew what the answer actually is. Is this a result that’s accepted by the mathematical community? Or are we just being lied to by the internet?
I think Calculus 2 (which includes a fairly detailed coverage of Taylor series) is a great moment to be asking this question; here was my response:
The very short answer to your question is, no, it is not really true that the sum of the positive integers is equal to −1/12. But there is an interesting piece of mathematics that lies behind that equation, and it’s not quite the total nonsense it appears on first glance. I recommend the following blog post by a professor at Utah:
Here is my supporting commentary to go along with it. The mathematical context called is the field called “complex analysis” — it’s essentially what you get if you do calculus, but you use the complex numbers as your basic set of numbers instead of the real numbers. Many things work out exactly the same; for example, the definition of the derivative is exactly the same, except that the little h going to 0 is now a complex number instead of a real number. And one can still build up all the machinery of Taylor series in this context. Now, we’ve seen that sometimes Taylor series converge only in some cases. For example, we know
but only for certain values of x (namely |x| < 1), even though the function on the right is defined in other situations (for any x other than 1). So, the equation you’re asking about is similar to
with the “proof” that you put x=2 into that formula for geometric series above. This is not right (we’re outside the interval of convergence, and the series on the left diverges), but it’s possible to understand what is meant by the equation. The 1+2+3+4+… example is similar, but with a more sophisticated series.
Q: What do you call it when in early January you use an exact sequence to describe the structure of a module?
My parents (originally from Michigan and Maine; well-educated native speakers of American English) and I agree that the comic above is sufficiently incomprehensible as to be not funny, but also that it’s written in non-standard English: the phrase “at all” at the end is very odd to our ears. Upon seeing it, I immediately wondered if the author is Minnesotan (as it turns out, he’s an Arizona native), because the strange use of “at all” in interrogatives is common here, too. Usually, I’ve run across it in the context of a cashier or waitperson asking the question “Would you like a receipt at all?” or “Can I get you the check at all?” Grammatically, it seems to be functioning as a generic question word (that is, if I were to translate from the local dialect into my own, I would simply remove the phrase and leave a question mark), but I haven’t tried to discern whether it also conveys a subtlety of meaning (nor do I know enough linguistics to know what sort of thing to be looking for here).
Maybe it is worth noting that there are contexts in which I would use “at all” as part of a question: for example, “Can I get you anything at all?” sounds correct and natural.
It was -15F or so when we flew in yesterday, with wind chills in the -40 to -60 range. This beats by several degrees the coldest day since I arrived in August 2012, and in fact led to school closures state-wide, and national news coverage. So, I had a lot of discussions with friends and family in NY about the weather. Eventually, I came up with a good way to explain the situation: in NYC, one goes outside in boots, gloves and a proper coat; sometimes it gets fairly cold (say, into the low 20s or high teens) and one thinks, “gosh, I would be more comfortable with a hat, a scarf and perhaps long underwear.” In Minneapolis, when I get ready in the morning I put on long underwear, a scarf, gloves, boots, a hat and a proper coat without checking the weather report first.
I follow the Times in using a completely inappropriate title for my piece; theirs is here. A friend asked for my feelings about it. When I finally read it, here were my scattered comments, beyond the terrible title:
- I thought the piece was poorly argued. For example, early in the article we see the claim that “within five years, there will be 2.4 million STEM job openings” used to support the idea that “the need for [STEM] workers is soaring,” but this is obviously fallacious — in the next five years, there will be several thousand academic job openings in math departments, but this big number is substantially smaller than the number of people with Ph.D.s in mathematics who are likely to be seeking those jobs. Fractions have denominators. Subsequently, “[American students] ranked 30th among students in 65 nations in math” is used to support the view that “[they] need vastly improved skills in math and science,” and this is just a non sequitur.
- Otherwise, it seemed like a grab-bag of not-unreasonable policy suggestions that don’t seem to have much to do with each other or with the shortage/crisis scaremongering.
- I thought http://opinionator.blogs.nytimes.com/…/the-real…/ was a better piece — substantial portions of it are not really specific to the humanities at all but apply to any search for pure knowledge.